ABC_EPP

8/9/2019 ABC_EPP

http://slidepdf.com/reader/full/abcepp 1/9

Applications of an annihilating boundary operator to water wave diffraction by a vertical circular cylinder 1

E E nvironmentnvironment P P rotectionrotection P P rogramrogram

ADAM SZYMAŃSKIADAM SZYMAŃSKIKierzkowo 22A, 84-210 ChoczewoKierzkowo 22A, 84-210 Choczewo

PolandPoland [email protected]

[email protected][email protected]

AApplications Of An Annihilatingpplications Of An Annihilating

Boundary Operator To Water WaveBoundary Operator To Water Wave Diffraction By A Vertical Circular Cylinder Diffraction By A Vertical Circular Cylinder

An annihilating boundary operator is introduced to examine the effects of water wave interaction

with a circular cylinder characterized by the dissipative or nondissipative surface properties.

Comparisons with models proposed by Garrett 1971

, Black et al.1971

, and experimental data of

Massel and M azenrieder 1983are also presented.

TECHNICAL REPORTECHNICAL REPORT2010

Fig 1a: Schematic illustration of the model geometry

mailto:[email protected]






8/9/2019 ABC_EPP



1. Introduction

TThis report deals with the diffraction of a time-harmonic water wave by a three-

dimensional vertical circular cylinder which is characterized by the dissipative surface

properties. The problem is investigated by means of the linear theory of water waves. Weemploy an annihilating boundary operator for problems of wave interaction with different

structures used in practical applications. Our results are compared with solutions for a semi-

immersed circular cylinder (cf., Garrett 1971), and a submerged circular cylinder protruding from

sea bottom (cf., Black et al.1971). The experimental data reported by Massel and Manzenrieder

1983are presented. We also treat wave interactions with a full-depth-draft circular cylinder having

dissipative surface.

2. An annihilating boundary operator

TThe velocity potential for a complex-valued incident small-amplitude water wave of

angular frequency ω can be written as,

φin = -igA ω-1 exp(-iωt) cosh[k(z+h)][cosh(kh)]-1 exp[ikrcos(λ)] (1)

where the wave number k is related to the angular frequency ω through the dispersion relation;

ω2 – gk tanh(kh) = 0, (1a)

and A is the wave amplitude, h is the water depth, g is the acceleration due to gravity, i = 0 + i1, (λ, r, z) are the cylindrical coordinates (cf., Fig. 1a and 1b), t denotes the time-variable.

For a circular cylinder of radius a it is easy to show that,

L φin = 0, r = a (2)

Fig 1b: The polar coordinates

and an incident wave

8/9/2019 ABC_EPP



where

L ≡ [ ∂/∂r - ik cos(λ)]. (2a)

The differential operator on the left-hand side of Eq. 2 annihilates incident wave φin, so it may be

called annihilating boundary operator. The operator (2a) was also considered by Higdon 1986

. It isan operator that annihilates in different directions, as a generalization of the Engquist-Majda

operator 1977. A more general form of L may be written as follows,

L[Z] ≡ [ ∂/∂r - ik Z cos(λ)] (3)

where Z is a complex quantity and is called the surface impedance.

3. The boundary value problem and its solution

TThe irrotational flow of an inviscid, incompressible fluid, for the geometry shown in Fig.

1a, is assumed to be described by the velocity potential φ. Additionally, it is assumed that sea

bottom is impervious and the excitation is provided by a train of simple harmonic waves φin of

small amplitude A. The total velocity potential φ is decomposed into an incident φin and a

scattered wave potential φsc according to

φ = φin + φsc. (4)

The total wave field is completely specified once φsc is known. According to the above

assumption the boundary value problem for φsc is defined as follows,

▽2φsc = 0 -h ≤ z ≤ 0, r ≥ a (5)

[ ∂/∂z – ω2/g] φsc = 0 z = 0, r ≥ a (6)

L[Z = Z1] φsc = - L[Z = Z1] φin -ht ≤ z ≤ 0, r = a (7a)

L[Z = Z2] φsc = - L[Z = Z2] φin -h b ≤ z < ht, r = a (7b)

L[Z = Z3] φsc = - L[Z = Z3] φin -h ≤ z < h b, r = a (7c)

∂/∂z φsc = 0 z = -h r ≥ a (8)

lim r 1/2

( ∂/∂r -ik ) φsc = 0. (9)r → ∞

The complex-valued scattered wave potential φsc can be expressed in the following form,

∞ ∞

φsc = -igA ω-1 exp(-iωt) ∑ ∑ Cns K n(αs r) cos[αs (z + h)] [cos(αs h)]-1exp[in( λ+π/2)] (10)

n = - ∞ s = 0

where K n(αs r) is the modified Bessel function of the second kind, α0 = -ik, and the eigenvalues

αs for s = 1, 2, 3,..., are determined by

8/9/2019 ABC_EPP



ω2 + gαs tan(αs h) = 0, (11)

for the evanescent modes. The function given by the relation (10) satisfies the Laplace equation

(5), the combined free-surface condition (6), the condition at the sea bottom (8) and the

Sommerfeld radiation condition (9) at infinity. The remaining boundary conditions to be satisfied

are Eqs. (7a,b,c), and determine the values of the coefficients Cns. For calculating thecoefficients Cns we have used the Galerkin method [e.g., Kantorovich and Krylov 1958].

4. Comparison with existing results

4.1 Wave forces on a circular dock (Garrett 1971)

UUsing the model proposed we have calculated the horizontal force Fx on semi-immersed

circular cylinder for the following geometrical quantities: ht/h = 0, (h – h b)/a = 1, h/a = 1.5, Z2 =

0 + i0 and Z3 = 1 + i0. The results obtained are presented in Fig. 2, where ρ denotes the densityof water.

4.2 Wave forces on a circular cylinder protruding from the sea bottom ( Black et al. 1971)

FFor a cylinder protruding from the sea bottom the horizontal force F x is presented in Fig.

3 for the following set of parameters: h t/h = 0.5, h b/h = 1, a/h = 1, Z1 = 1 + i0, Z2 = 0 + i0. Our

model is compared with the results of Black et al. 1971.

Fig.2 Horizontal force on a circular dock

8/9/2019 ABC_EPP



4.3 The pressure around the circumference of a submerged cylinder ( Massel and Manzenrieder 1983)

TThe pressure p around the circumference (-z/h = 0.939) of a submerged cylinder is

plotted in Fig. 4 for: ht/h = 0.9 h b/h = 0.978, a/h = 0.508, kh = 1.044, Z1 = 1 + i0, Z2 = 0 + i0 and

Z3 = 1 + i0. Fig. 4 also shows experimental pressure data reported by Massel and Manzenrieder 1983

.

Fig.3 Horizontal force on a cylinder protruding from the bottom

Fig.4 Pressure around the circumference of a submerged cylinder

8/9/2019 ABC_EPP



Next, a submerged cylinder on a rubble-mound base is considered. The pressure around the

circumference (-z/h = 0.836) of this structure, having partially dissipative properties, where h t/h

= 0.676, h b/h = 0.764, a/h = 0.572, kh = 0.779, Z1 = 1 + i0, Z2 = 0 + i0, Z3 = 0.2 + i0 is presented

in Fig. 5 with the experimental data of Massel and Manzenrieder 1983

.

5. The free-surface elevation around the full-depth-draft cylinder

TThe free-surface elevation for the total wave field around the full-depth-draft cylinder for

t = t0 is shown in Figs. 6a, b, c, d, e for the surface impedance coefficient Z1 = Z2 = Z3 = Z equal

to 0 + i0, 0.5 +i0, 0.9 + i0, 0.99 + i0 and 0.9999 + i0, respectively.

Fig.5 Pressure around the circumference of a submerged cylinder

Fig.6a The free-surface elevation for ka = 8 and Z = 0 + i0

8/9/2019 ABC_EPP



Fig.6b The free-surface elevation for ka = 8 and Z = 0.5 + i0

Fig.6c The free-surface elevation for ka = 8 and Z = 0.9 + i0

8/9/2019 ABC_EPP



Note that Fig. 6a shows the limiting case of MacCamy and Fuchs 1954

. The impervious surface of nondissipative properties is recovered from (3) by assuming Z = 0 + i0, while Z = 1 + i0 results

in a boundary condition of undisturbed plane incident wave property ( cf., Fig. 6e, Z = 0.9999

+i0). Thus, the vertical circular cylinder surrounded by a coating consisting of a meta-material

Fig. 6d The free-surface elevation for ka = 8 and Z = 0.99 + i0

Fig.6e The free-surface elevation for ka = 8 and

Z = 0.9999 + i0

8/9/2019 ABC_EPP



might become invisible for the time-harmonic water waves of small amplitude. This formal

transparency has been used to approximate the natural boundary conditions used by Garrett 1971

.

Additionally, the model described above suggests the reasonable approximations to the exact

solutions of the so-called transmission problems, and does not require the laborious matching of

eigenfunction expansions.

6. Summary and Conclusions

TThis report presents the applications of an annihilating boundary operator for computing

wave forces on offshore structures. Examples studied include the full-depth-draft vertical circular

cylinder, the submerged circular cylinder protruding from the sea bottom and the well-known

Garrett's dock problem for which the solutions based on the impedance boundary conditions are

compared against analytical and experimental results. The examples presented above and their

comparison with the exact solutions validate the use of the impedance boundary conditions to

study waves diffraction and, in particular, points out the simplicity of the method proposed.

7. References

Black, J.L., Mei, C.C, Bray, M.G.G., 1971

Radiation and scattering of water waves by rigid bodies, J. Fluid

Mech., 46, 151-164.

Engquist, B., Maja, A.,1977

Absorbing boundary conditions for the numerical simulation of

waves, Mathematics of Computations, 21(139), 629-651.

Garrett, C.J.R., 1971

Wave force on circular dock, J. Fluid Mech., 46, 129-139.

Higdon, R.,L., 1986

Absorbing boundary conditions for the difference approximations

to the multi-dimensional wave equation, Mathematics of

Computations, 176, 437-459.

Kantorovich, L.V., Krylov, V.I., 1958

Approximate methods of higher analysis, P.Noordhoff Ltd.,

Groningen, The Netherlands.

MacCamy, R.C., Fuchs, R.A., 1954

Wave forces on piles: A diffraction theory, Tech. Memo., no. 69,

U.S. Beach Erosion Board, U.S. Army Corps of Engineers,

Washington, D.C.

Massel, S.R., Manzenrieder, H., 1983

Scattering of surface waves by submerged cylindrical body with porous screen, Lehrstuhl für Hydromechanik und

Küstenwasserbau, Leichtweiss-Institut, Technische Universität

Braunschweig, report no. 562

Documents

ABC_EPP